Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

插值曲线区域控制的加权有理插值方法

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题,文中利用分母为线性的有理三次插值样条和仅基于函数值的有理三次插值样条构造了一种加权有理三次插值样条,由于这种有理三次插值样条中含有新的参数,给约束控制带来了方便,给出了将插值曲线约束于给定的折线、二次曲线之上（下）或之间的条件,最后给出了数值例子。     

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R³ is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.     

Minkowski isoperimetric-hodograph curves

General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case of rational general offsets, namely, Minkowski isoperimetric-hodograph curves. Differential geometric topics in the Minkowski plane, including the notion of normality, Frenet frame, Serret–Frenet equations, involutes and evolutes are introduced. These lead to an elegant process from which an explicit parametric representation of the general offset curves is derived. Using the duality between indicatrix and isoperimetrix and between involutes and evolutes, rational curves with rational general offsets are characterized. The dual Bézier notion is invoked to characterize the control structure of Minkowski isoperimetric-hodograph curves. This characterization empowers the constructive process of freeform curve design involving offsetting techniques.     

带有给定切线多边形的曲率连续的有理二次样条曲线

本文论述了与给定切线多边形相切的有理二次Bézier曲线,构造曲线是曲率连续的,具有局部可调性,且对切线多边形是保形的;跟三次（四次）Bézier曲线或B样条曲线方法相比,具有切点的变动范围更大、曲线次数低、结构简单、计算量少、显示更快的特点。最后,通过实例加以说明。     

Detecting symmetries of rational plane and space curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.     

Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves

An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. The result is an angular function θ(t) of the curve parameter, comprising in general both rational and logarithmic terms, that specifies the orientation of the rotation-minimizing frame relative to the Frenet frame. For PH cubics and quintics, the solution employs only arithmetic operations on the curve coefficients and some complex square and cube root extractions. Moreover, the generalization to PH curves of arbitrary order entails only standard polynomial algorithms (i.e., arithmetic, greatest common divisors, and resultants), solution of a linear system, and a minimal element of polynomial root-solving. Rotation-minimizing frames are employed in computer animation, the construction of swept surfaces, and in robotics applications where the axis of a tool or probe should remain tangential to a given spatial path while minimizing changes of orientation about this axis.     

可调的类三次Bzier三角曲线

本文给出了一种基于函数1、sinu、cosu和sin2u的可调类三次参数曲线,由四个顶点控制的曲线不仅具有类似于三次Bzier曲线的诸多性质,而且其形状可由一个参数进行调节,使得该曲线具有更强的表现能力。为便于自由曲线的设计,还讨论了两段曲线的拼接条件。结果表明,该曲线在拼接方面比三次Bzier曲线具有优越性,在适当选取形状参数时,两条曲线可在连接点处达到C3拼接,其拼接条件也比三次Bzier曲线简单得多,因此该曲线更适用于曲线造型。另外,该曲线无需有理形式即可精确地表示圆、椭圆、抛物线等二次曲线,方便实际应用。     

一种有理三次样条的约束插值

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题。构造了一种仅依赖于函数值的分母为二次的有理三次插值样条,是[C1]连续的,使用起来较方便,并含有参数,具有较好的可约束控制性质。研究了该样条曲线的区域控制问题,讨论了该插值曲线约束于给定折线二次曲线上（下）方或之间的条件,并给出了数值算例。所给约束条件容易满足,便于使用。     

有理q-Bernstein-Bézier曲线的构造及其应用

有理Bernstein-Bézier曲线在计算机辅助设计和计算机图形学上具有广泛的应用。在研究了经典的Bernstein-Bézier曲线及de Casteljau算法的基础上,结合q-Bernstein多项式,给出了有理q-Bernstein-Bézier曲线的构造方法、性质和计算有理曲线的de Casteljau算法,并讨论了曲线的细分和升阶的方法,通过改变q的取值,可以获得有理曲线族,在曲线造型上具有较强的灵活性。最后通过表示圆锥曲线和数字图像插值证明有理q-Bernstein-Bézier曲线的推广是有效的。     

基于三次PH 曲线误差可控代数曲线等距线逼近算法

论文提出一种用三次PH 曲线逼近代数曲线的方法及其误差分析。使用该 方法,给出一种用PH 曲线的等距线来逼近原来代数曲线等距线的算法。逼近曲线保持了原 曲线的一些重要几何性质,如单调性、凹凸性、G1 连续性等。数值实验表明,该算法提供 了代数曲线近似参数化的一条有效途径。并在此基础上提出了一种计算代数曲线等距线的有 理参数表示的新方法。     

The method of resolvents: A technique for the implicitization, inversion, and intersection of non-planar, parametric, rational cubic curves

Let P(t) be a non-planar, parametric, rational cubic curve. The method of resolvents is applied to: (1) construct three quadric surfaces whose intersection is equal to P(t) (implicitization); (2) solve for the parameter t as the ratio of two linear expressions in the coordinates x, y, z (inversion). The results of these two operations are then applied to construct an optimal, robust, intersection algorithm for any two non-planar rational cubic curves, and it is shown that two such curves can intersect in at most five points. Specializations of these results for non-planar, integral, cubic curves are derived, and extensions of these techniques to non-planar, rational cubic, Bézier curves are also discussed.     

The rational cubic Bézier representation of conics

The rational cubic Bézier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bézier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bézier vertices.     

一类代数-三角函数表示的空间PH 曲线及其应用

在代数-三角函数空间Ω=span{1,θ ···, θm+1, sinθ, cosθ, θsinθ, ···, θn cosθ}定义了一类 空间曲线。通过选取合适的积分核函数,该曲线在xy-平面上的投影具有内蕴表示或整条曲线是 PH 曲线。曲线的笛卡尔坐标可由预定义的核函数通过积分计算得到。此外,给出了不同核函数 表示的积分曲线的Hermite 插值算法。对给定的边界条件,积分核函数系数可通过求解方程组 得到。最后,利用PH 曲线设计了一族标架,并用于构造有理形式的扫掠曲面。实验表明,分 片定义的扫掠曲面在脊线处G1 连续,在其余连接处达到近似G1 连续。     

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation.When all th knot spans are identical,the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves.If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.     

曲率连续的有理二次样条插值的一种优化方法

人们通常用有理三次曲线样条来构造整体曲率连续的曲线.提出利用有理二次样条曲线插值整体曲率连续的曲线的一种方法.首先导出了两相邻二次曲线段间曲率连续的拼接条件,然后提出了求解平面上一个闭的点列中每一点处的切线的最优算法.最后给出了闭曲线插值的一些实例以检验方法的有效性.     

高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。     

三次Ball曲线的两种新扩展

给出了次数分别为3和4的含参数的多项式基,它们都是三次Ball曲线基函数的扩展。基于这两组基函数定义了两类带形状参数的多项式曲线,新曲线不仅具有三次Ball曲线的特征,而且具有形状可调性和比三次Ball曲线更好的逼近性。通过分析新曲线与Bézier曲线之间的关系,得出了形状参数的几何意义,并给出了新曲线的几何作图法。     

Bézier曲线的新扩展

给出2组含有2个形状控制参数 的四次、五次多项式基函数,其分别是三次、四次Bernstein基函数的扩展。分析2组基的性质,定义带 的2类多项式曲线：三次E-Bézier曲线和四次E-Bézier曲线,其具有三次或四次Bézier曲线的特性、形状可调性和更好的逼近性。当 时,2类曲线分别退化为三次、四次Bézier曲线。给出2个扩展曲面的定义。实例表明,定义的曲线为曲线/曲面的设计提供了一种有效的方法。